Standard deviation for SNG's and sample size for CASH games

[jay_shark]

I'm not sure if anyone has written extensively on this topic in regard to heads up games but I'll try to explain things as simple as possible .

Your standard deviation for heads up sng's is a function of your win rate . If you assume that your win rate is ~60% , then your standard deviation will remain fixed for any player who shares the same win rate . A simple calculation proceeds as follows :

Var(x)=E(x^2)-E(x)^2 where E(x) is your win rate(or mean) for a random variable x . For this particular case , we regard the variable x as +1 for when we win 1 unit and -1.05 for when we lose one buyin which also includes the rake .

So assuming you win 60% of the time , your s.d is simply the square root of your variance or var(x) .

E(x^2)=1^2*0.6 +(-1.05)^2*.4
E(x^2)= 1.041

Also E(x)=1*0.6-1*0.4 -0.05
E(x)=0.15

Var(x)=1.041-0.15^2
Var(x)=1.0185

S.D(x)=sqrt(1.0185)~ 1.0092

This means that if your win rate is 60% , then your s.d is approximately equivalent to 1 buyin , no matter what !

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Our standard deviation for cash games may be different for two players who share the same win rate . As I explained earlier , this is not the case for sng's .

To compute our standard deviation for cash games , we need at least 30 cash game sessions to ascertain that our s.d converges to a steady number . We need not even know our win rate which is why the central limit theorem is so very useful in situations like this . Using the numbers that Jakeduke provided , I will calculate the number of hands needed to determine with 95% confidence, our win rate interval .

Lets say after 50k hands , our win rate is (10 bb)/100 hands and bb is not to be mistaken for big bets .

Our standard deviation , using jakeduke's numbers is ~ 11 big blinds/100 hands .

xbar is our sample mean (10 bb/100 hands)
z= our confidence level which is approximately 2 s.d's above and below the mean .
sigma bar is our sample standard deviation .

In this case , sigma bar is 11/sqrt(500)~ 0.4919 per 100 hands .

10 +- 2*0.4919 Which means that we are 95% confident that Jakeduke's win rate lies between 9.0162 to 10.98 BB's per 100 hands .

Using z=3 , we are about 99% confident that Jake's win rate lies between 10 +- 3*0.4919 or between 8.52 to 11.47 BB's per 100 hands .

[derosnec]

but per how many hu sngs?

it's not 1 per 1000 games.

does my question make sense?

[hra146]

You lost me right after "simple math".


But I dont even understand the concept of SD so I guess... yeah.

[daveT]

[ QUOTE ]
You lost me right after "simple math".




[/ QUOTE ]

[jay_shark]

Let me add more to this discussion .

Let x be the number of hands needed to be within 1ptbb/100 of your hypothetical win rate . So if we believe that we win 10bb/100 hands and our standard deviation is 110bb/100 hands , then the number of hands needed at 2 standard deviations is :

10+-2*(110/sqrt(x/100))
So we want 2*(110/sqrt(x/100)) =1
sqrt(x/100)=+-2*110
sqrtx=+-2*110*10
x=4 840 000

This means that even after 4 840 000 hands , you are 95% confident that you're within 1 bb/100 of your true win rate . This is one of the reasons why nobody really knows their true win rate because they usually change limits before they can draw any conclusions from it .

[jay_shark]

One correction .

Instead of S.D =11BB/100 hands I meant to put S.D=110bb's/100 hands . No wonder why things weren't adding up .

Jake's true s.d in terms of big blinds is 110bb/100 hands .

Sigma bar becomes 110/sqrt500 = 4.919/100 hands .

Our confidence interval becomes :

10+-2*4.919

and so we're 95% confident that his true win rate lies between 0.162 to 19.838 . All this tells us is that we're pretty confident that he's a winner.

This is one of the reasons why they say 50k hands is sufficient to determine that you're a profitable poker player .

[jay_shark]

[ QUOTE ]
but per how many hu sngs?

it's not 1 per 1000 games.

does my question make sense?

[/ QUOTE ]

For each 1 sng , your standard deviation is +1 unit from your mean . For a 9 player sng , your standard deviation will be close to 1.7 for each 1 sng .

[jay_shark]

Tnixon , here is the day you've been waiting for [img]/images/graemlins/smile.gif[/img]

Jakes s.d for NL100 is 110 bb's/100 hands .
Lets make a hypothetical assumption that an average sng lasts 30 hands . So playing a $100+5 buyin sng at a 60% win rate is equivalent to a s.d of about $100/30 hands which is about THREE times higher than your variance playing in cash games !!

[jay_shark]

One other minor correction .

If we include rakes into this discussion for an sng , then this means (as a 60% winner) that you will win 0.95 units , 60% of the time , and lose 1.05 units , 40% of the time .

So for my original variance calculation , we should replace +1 with +0.95 but this doesn't change things much .

E(x^2) = 0.95^2*0.6 + (-1.05)^2*0.4
E(x^2)=0.9825

var(x)= 0.9825 - 0.15^2
var(x)=0.96
s.d(x)=0.97979797 or almost 1 buyin per sng .

[TNixon]

[ QUOTE ]
Tnixon , here is the day you've been waiting for

[/ QUOTE ]

The day I've been waiting for? You mean the day where you run through the same calculations that I already showed in another thread (although to be fair, my numbers were off, and showed SNGs being lower variance than they really are, because I incorrectly put the winrate inside the squared terms instead of outside, where it belonged), and ended up basically restating the same conclusion that I already came to in another post, without any acknowledgement whatsoever that it even existed, even though I know for a fact that you at least browsed it?

Wow. You're right. I've totally been waiting for that. In fact, my face is blue, I've been holding my breath for so long.